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Auslander algebras and initial seeds for cluster algebras
Author(s) -
Geiß Christof,
Leclerc Bernard,
Schröer Jan
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm017
Subject(s) - quiver , indecomposable module , mathematics , injective function , lambda , combinatorics , unipotent , cluster algebra , pure mathematics , group (periodic table) , physics , statistical physics , quantum mechanics , ising model , optics
Let Q be a Dynkin quiver and Π the corresponding set of positive roots. For the preprojective algebra Λ associated to Q , a rigid Λ‐module I Q is produced with r = |Π| pairwise non‐isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of k Q to Λ. If N is a maximal unipotent subgroup of a complex simply connected simple Lie group of type | Q |, then the coordinate ring ℂ[ N ] is an upper cluster algebra. It is shown that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of I Q coincide with certain generalized minors which form an initial cluster for ℂ[ N ] and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of End Λ ( I Q ). Finally, the fact that the categories of injective modules over Λ and over its covering Λ̃ are triangulated is exploited in order to show several interesting identities in the respective stable module categories.